Eigenvector-free solutions to AXÂ =Â B with PXÂ =Â XP and XHÂ =Â sX constraints

The matrix equation AX = B with PX = XP and XH = sX constraints is considered, where P is a given Hermitian involutory matrix and s = ±1. By an eigenvalue decomposition of P, we equivalently transform the constrained problem to two well-known constrained problems and represent the solutions in terms of the eigenvectors of P. Using Moore-Penrose generalized inverses of the products generated by matrices A, B and P, the involved eigenvectors can be released and eigenvector-free formulas of the general solutions are presented. Similar strategy is applied to the equations AX = B, XC = D with the same constraints.